Jordan algebras. We recall that an abstract Jordan algebra \(\mathcal {J}\)  is an algebra over a field with a product that satisfies \(a \circ b = b \circ a\)  and \((ab)(aa) = (a(b(aa))\)  for all \(a,b \in \mathcal {J}\) .For an extensive study of Jordan algebras, see [1]}.. Given an associative algebra \(\mathcal {A}\)  one obtains a Jordan algebra \(\mathcal {J}(\mathcal {A})\)  by symmetrisation. If \(\mathcal {J}= \mathcal {J}(\mathcal {A})\)  for an associative algebra \(\mathcal {A}\) , then the Jordan algebra is called special, otherwise it is called exceptional.The prototypical exceptional Jordan algebra is the so-called Albert algebra \(H_3(\mathbb {O})\)  [2]}.
In particular, every \(C^*\) - (and von Neumann)Recall that a \(C^*\) -algebra is an involutive Banach algebra (closed in norm) satisfying the defining \(C^*\) -property, \(||x^*x|| = ||x||^2\) . A von Neumann algebra is a \(C^*\) -algebra closed in the weak operator topology. algebra defines a JB(W) algebra: a JB(W) algebra is a (weakly closed) Jordan algebra that is also a Banach space (\(||a \circ b|| \le ||a||\cdot ||b||\) ) such that \(||a^2|| = ||a||^2 \le ||a^2 + b^2||\) . For simplicity, here we only consider matrix algebras over the complex numbers, \(\mathcal {A}= M_n(\) , \(n \in \mathbb {N}\) .
In this case, the set of Hermitian matrices \(H_n(\)  under the anti-commutator \(\lbrace a,b\rbrace  := ab + ba\)  defines a real Jordan algebra \(\mathcal {J}(\mathcal {A})_\mathrm {sa} := (H_n(,\lbrace \cdot ,\cdot \rbrace )\) . We denote its complexification by \(\mathcal {J}(\mathcal {A}) = \mathcal {J}(\mathcal {A})_\mathrm {sa} +i\mathcal {J}(\mathcal {A})_\mathrm {sa}\) .
